Infinite quotient rule?

Calculus Level 3

F(x)=xx+xx+xx+x...\large F(x) = \frac{x}{x + \frac{x}{x + \frac{x}{x + \frac{x }{._{._.}}} } }

If the value of the first derivative of F(x)F(x) at x=1x = 1 can be expressed as abbc \frac{a\sqrt b - b}{c} for integers a,b,a,b, and cc, find the value of a+b+c.a+b+c.

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